Advanced 8-day course
Introduction to Smooth Manifolds and Observables
November 28 - December 8, 2006, Lublin, Poland
Lecturer: Giovanni Moreno

The course "Introduction to Smooth Manifolds and Observables" was intended as an introduction to the standard B1 course usually held at Diffiety Schools. No exercises were proposed to the students, and it covered the first half of the B1 program. Diplomas of participation were delivered to all the participants.
People who followed this course and are willing to discover more about Jet Nestruev might participate to some of the next editions of the Diffiety School, but they must be aware that the B1 course is wider and that several exercises are given.

Course schedule and arguments

Lecture I - Tuesday, November 28, 12:30-15:00
Introduction to the philosophy of Jet Nestruev. Advertisement of the Diffiety Schools.
Observability principle. Boolean algebras. Looking beyond the observable world: gauge theories and quantum mechanics.
Measuring devices. Set of the states.
Definition of algebra.
Lecture II - Wednesday, November 29, 17:30-20:00
Basic topology. Neighborhood systems and continuity. Weak topology.
On the smoothness of physical phenomena.
Introduction to charts as special observables. Smoothness of a curve in the set of the states.
Lecture III - Thursday, November 30, 12:30-15:00
Charts, atlases and smooth manifolds.
On the concept of smoothness on a smooth manifold.
Difference between local and global properties.
Constructing a smooth atlas on the Grassmann manifold.
A remarkable smooth map: the Gauss map on a surface.
Lecture IV - Friday, December 1, 08:00-10:30
Smooth maps, curves and functions on smooth manifolds.
Categories and functors.
Functor C∞.
C∞-closure of the algebra C∞(M).
Special seminar for professors and doctors - Monday, December 4, 12:00-14:00
General introductory facts about the geometry of nonlinear PDEs. Explaining how the observability mechanism leads naturally to the main constructions of Secondary Calculus.
Lecture V - Tuesday, December 5, 12:30-15:00
Bump functions. Hadamard lemma. A function with compact level surfaces.
Points as algebras homomorphisms.Geometric algebras, with an example of a non-geometric one.
Lecture VI - Wednesday, December 6, 16:40-19:10
The ideal of ghosts. Making an algebra geometric.Spectral theorem for an Euclidean open domain.
Complete algebras, with an example of a non-complete one.
Completeness of the smooth functions algebra.Definition of smooth algebras.
Definition of smooth maps between the spectra of algebras.
Proving that the spectrum of a smooth algebra is a smooth manifold, and that the algebra coincides with its smooth functions algebra.
Lecture VII - Thursday, December 7, 14:45-16:30
Introduction to dynamics of a physical system. Historical remarks on the Leibnitz rule and the birth of infinitesimal calculus.
Understanding the linearity of a map as the property of commuting with the the operations.
Definition of modules by introducing the multiplication as an algebras homomorphism.
Introducing differential operators of order less than 1 as a generalization of modules homomorphisms.
Derivations. Tangent space.
Theorem of the tangent vector on ℝn.
Lecture VIII - Friday, December 8, 08:00-10:30
Locality of tangent vectors.Differential of a smooth map.
Theorem of the tangent vector on any manifold.
Examples of algebras admitting no dynamics: C0(ℝ) and the boolean algebra.